Econ 510a (second half)Yale UniversityFall 2007Prof. Tony SmithFINAL EXAMINATIONThis is a closed-book and closed-notes exam. You have three (3) hours to complete the exam.There are four (4) questions on the exam for a total of 100 points. The points allocated toeach part of each question are indicated below. Please put the answer to each question ina different blue book. To receive full credit, you must provide convincing explanations tosupport your answers. Please write as neatly as possible.1. A consumer seeks to maximize her lifetime utility of consumption and leisure, whichis given by:P∞t=0βtu(ct, `t), where ctis consumption in period t and `tis leisure inperiod t. The felicity function u is strictly increasing and strictly concave in both ofits arguments.The consumer is endowed with one unit of time in each period, which she allocatesbetween leisure and work. The consumer’s labor income in period t equals whtnt, wherew is the (time-invariant) wage per unit of human capital, htis the the consumer’s levelof human capital in period t, and ntis the amount of time that the consumer spendsworking in period t. Human capital accumulates over time according to:ht+1= (1 − δ)ht+ f(ht, nt),where f is strictly increasing in both arguments , strictly concave in its first argument,and satisfies: f(0, n) = f (h, 0) = f(0, 0) = 0. In other words, human capital depreci-ates at rate δ, but the consumer can invest in human capital by working. The consumerhas human capital equal to h0> 0 in period 0. To keep things simple, s uppose thatthe consumer does not participate in asset markets and instead simply consumes herentire labor income in every period.(a) [8 points] Formulate the consumer’s dynamic optimization problem as a dynamicprogramming problem. That is, display the consumer’s Bellman equation andidentify clearly the control (or choice) variable(s) and the state variable(s).(b) [9 points] Find an equation that determines (implicitly) the steady-state level ofhuman capital.1(c) [5 points] Suppose instead that f is linear: f (ht, nt) = g(nt)ht, where g is strictlyincreasing and satisfies g(0) = 0. Do you think that the consumer’s optimal pathfor human capital converges to a steady state in this case? Explain.2. Consider a version of a “Lucas tree” economy in which there are two types of trees.Both types of trees are perfectly durable; a type-i tree (i = 1, 2) yields a randomamount of dividends equal to ditin period t. Assume that {d1t}∞t=0and {d2t}∞t=0arei.i.d. sequences of random variables and that d1tand d2sare statistically independentfor all t and s. In addition, for i = 1, 2, assume that ditequals dLwith probability πiand equals dH> dLwith probability 1 − πi.The economy is populated by a continuum (of measure one) of identical consumerswith preferences over consumption streams given by:E0∞Xt=0βtlog(ct),where ctis consumption in period t. In period 0, each consumer owns one tree of eachtype. Dividends are non-storable and are the only source of consumption goods. Thereare competitive markets in which consumers can buy and sell both types of trees.(a) [5 points] Define a sequential competitive equilibrium in which the only assetsthat consumers trade are the two (types of) trees.(b) [9 points] Find an algebraic expression for the equilibrium price of a type-1 tree(measured in terms of today’s consumption goods), assuming that the dividendsof both types of trees are equal to dLtoday. Your expression should depend onlyon primitives (i.e., on the parameters describing preferences and technology).(c) [9 points] How many Arrow securities are there in this economy? Express theprices of these securities in terms of primitives.(d) [5 points] Use your answer from part (c) to find the price (expressed in terms oftoday’s consumption goods) of an asset that pays one unit of the consumptiongood in the next period if the dividends of the two trees (in the next pe riod) arenot equal to each other and pays zero otherwise.23. Consider a Diamond overlapping-generations economy in which there are two typesof consumers. In each period, a continuum of measure one of each type of consumeris born. Each type of consumer lives for two periods and supplies one unit of laborinelastically when young and zero units of labor when old. Each type of consumer isborn with zero assets. Type-i consumers, i = A, B, who are born in period t havepreferences given by: ui(c1t) + βui(c2,t+1), where c1tis consumption when young andc2,t+1is consumption when old. Let uA(c) = log(c) and uB(c) = (1 − σ)−1c1−σ, whereσ 6= 1. Let the economy’s aggregate production function take the form: y = kαn1−α,where y is aggregate output, k is the aggregate capital stock, and n is aggregate laborsupply. Assume that capital depreciates fully in one period.(a) [5 points] Find the savings decision rule of a young consumer of each type.(b) [9 points] Use your answer from part (a) to find an equation that determines(implicitly) the equilibrium law of motion of the aggregate capital stock.(c) [8 points] Under what conditions is the steady-state of this economy dynamicallyinefficient? Explain.4. Consider a neoclassical growth model populated by identical consumers with prefer-ences over consumption streams given by:P∞t=0βtu(ct), where ctis consumption inperiod t. Consumers do not value leisure and supply one unit of labor inelasticallyin every period. Firms have identical constant-returns-to-scale production functions.In every period, the government taxes labor income in order to subsidize investmentexpenditures. That is, the period-t budget constraint of a typical consumer reads:ct+ (1 + τ)xt= rtkt+ (1 − θt)wt,where rtis the period-t rental rate of capital, wtis the period-t wage rage, ktis theconsumer’s capital stock at the beginning of period t, xt≡ kt+1− (1 − δ)ktis theconsumer’s period-t investment expenditures, τ < 0 is the (time-invariant) subsidyrate on investment expenditures, and θt> 0 is the period-t tax on labor income. Thegovernment balances its budget in every period.(a) [9 points] Carefully define a recursive competitive equilibrium for this economy.(b) [7 points] Find a second-order diff erence equation that governs the evolution ofthe aggregate capital stock in equilibrium.3(c) [6 points] Is the steady-state aggregate capital stock higher or lower than it wouldbe if τ = 0? Explain.(d) [6 points] Is the competitive equilibrium allocation Pareto efficient when τ <